Dynamics of the trp Operon

Refs: Sántilan & Mackey, PNAS 98 (4), 1364-69 [2001]

I. Rahmim doctoral thesis, Columbia Univ., 1990

Alberts, MBOTC III, pp. 417-419, Lewin, Genes V, 435-7

AKH9

Overview

(Santillan)

Organization of the trp Operon*• 5 genes code for enzymes:

* Cluster of genes controlled by a single (?) feedback regulatorymechanism.

The five genes are transcribed as a single mRNA molecule, allowing their expression to

be controlled coordinately.

There is one promoter.

Within the promoter is an operator.

Tryptophan repressor can bind to operator and deny access to RNA polymerase.

Because repressor binding stops transcription the regulation mode is called negative control.

Repressor Terminology

Activation (binding) of repressor requires binding of two tryptophan molecules:

Multiple RNA polymerases act together on an operon; multiple ribosomes act together

on even partially formed mRNAs:

Attenuators… act to terminate transcription prematurely.

They act before or within the coding region

When they are non-functional, transcription continues past attenuation region

When they are functional, termination occurs

Tryptophan synthesis: chorismate l-tryptophan

(the 5 enzymes that make tryptophan)

trp E Anthranilate synthase

Phosphoribosyl anthranilae transferase

trp D

trp C Phosphoribosyl isomerase/indoleglycerol phosphate synthase

trp B Tryptophan synthetase

Tryptophan synthasetrp A

Tryptophan Operon Promoters:

p1 (strong): Active repressor binds to trp operator and overlaps p1 with physical exclusion of RNA polymerase

p2 (low efficiency): Within distal part of trpD, results in production of mRNAs C, B, and A.

Thus, production of trp E and D is differentiated from production of trp C, B, and A.

Additionally, deactivation and degradation of mRNA CBA is much slower than for mRNA ED.

Just before trpE transcription is regulated through attenuation. At low tryptophan levels, reduces by 8

Enzymatic Control of Tryptophan Levels:

Production: Anthranilate synthase inhibited up to 100% by tryptophan.

Interconversion: Tryptophanase (tna) catalyzes tryptophan-indole interconversion.

Transport Control of Tryptophan levels:

aro P product facilitates transport of aromatic amino acids.

mtr product facilitates tryptophan transport. (mtr is activated with phenylalanine present, tryptophan absent.)

Product of tyrR modulates expression of aroP and mtr.

Transport dynamics have not been studied – we don’t know the transporter kinetic coefficients.

Goodwin’s compartmental triad in bacterial gene expression:

,

,

,

[ ][ ] [ ],

[ ][ ] [ ],

[ ][ ] [ ]

(efficiency of feedback repression by product)

1=

1

mRNA r d mRNA

enzyme d enzyme

product d product

r

d mRNAk gene k mRNA

dtd enzyme

k mRNA k enzymedt

d productk enzyme k product

dt

product

K ,

(a negative-going Hill coefficient) D product

n

Y

X

0.00

0.25

0.50

0.75

1.00

0.00 1.00 2.00

Hill Functions: F, F+, F-

4

4 4

4

4 4

All of these can be written with the non-dimensional variable, /

, , , = 1 = 1= 1

x

Kx

x

x

x

x

Overall Model for trp Operon

About concentrations:

• Molecules per cell (with a 'standard' cell size). Intuitive but not usually measured.

• Standard chemical concentrations, e.g. moles/ml, etc. Normal measurement.

• Stochasticity issues. 50 molecules per cell of volume 1m3 ~ 0.1 M.

The Effect of Cell Expansion on Concentration

• Expanding (dividing) cultures lose concentration even without degradation –

• amount in one cell is unaffected until division, and is then suddenly halved.

• concentration decreases steadily up to, during, and after division.

• Intracellular heterogeneous kinetics are affected by concentration, not amount.

Allowing for Volume ExpansionWithout Expansion:

generationdc

dt rate

With expansion:

generationdc dV+c

dt rate

which can be written:

generation generationdc dVc

dt rate rate

d

d

d

V Vk c

V Vk cdt

V Vk cdt

ln

lnThe term is frequently designated as . Expansion

increases the apparent degradation rate. It can be the most

significant factor affecting metabolite concentrion during growt

dd V

V k cdt

d V

dt

h!

Cell Volume

-1where is the specific growth rate, min .

-15 60 (in liters) = 4 10cV e

mRNA Equations

1

,mRNA

1 2

,mRNA

[mRNA ]

( )[mRNA ]

[mRNA ]

( )[mRNA ]

p r a T

d ED

p r a T p T

d CBA

d EDv g

dtk ED

d CBAv g v g

dtk CBA

Repression:

1

1

r2

[apoR] [tryp]apoR+ tryp apoR tryp,

[apoR tryp]

[apoR tryp] [tryp]apoR tryp tryp holo ,

[holo R]

[holo R] [operator]holoR free operator bound repressor, K

[bound repressor]

"free" + "occupied" operato

r

r

K

R K

T

0

rs = g ;

apoR + apoR tryp + holoR + bound repressors = r ;

"occupied" operators = repressors bound to operators.trp trp

Intracellular Concentration* of Bound trp Repressors:

20

20 2

1

4

2where

(1 ) ,

and

[tryp]

T

T r

r

B B g r

B r g K A A

KA

__________________________________

* = molecules per cell/ (cell volume NA)

Finally, for repression:

conc'n of free operators

conc'n of occupied operators1

conc'n of bound repressors1

rT

T

T

trp

g

trp

g

trp

g

Attenuation

trp leader region: Transcription is stalled on the pause site until a ribosome arrives at base 27

How long does RNA polymerase take to move from the pause site to the potential termination site, i.e. what is T?

How far does the ribosome travel during time T?

Simulation:

RNA polymerase transcribes at an average rate of 42 nt/sec, but follows Poisson distribution.

Ribosome adds residues at a corresponding average rate of 14/sec, but dependent on amino acid availability. Rate is maximum at high, and zero at zero, concentration.

residues/sec.)[ ]

14 ([ ]ribosome

a

Av

K A

Assume tryptophan concentration controlling.

Monte Carlo Simulation Results:

3.0 [ ] /0.12

0.4 [ ] /a

aa

tryp K

tryp K

Other Coefficients Needed

• mRNA Degradation Rates (t1/2 values)

• Transcription Initiation Rates

Enzyme Balances

33 , 3

44 , 4

, 3 , 43 4

[mRNA ] ( )

[mRNA ] ( )

( ) ( );

[mRNA ] [mRNA ]

z d protein

z d protein

d protein d proteinz z

dzv ED k z

dtdz

v CBA k zdt

k z k zv v

ED CBA

Biosynthesis Pathway

max,

max,PRTas

,

,

e

[anth][ ]

[anth][ ] [anth]

K [anth]

and similarly for CdRP and tryp (with the addition of the

utilization term, , in the t

[tryp

ryp eq

]

ua

I ASase

I ASASase

PRTase

c

ase

dV ASase

dt

V PRTase

-

K

U

K

tion. The term in

represents inhibition of the first enzyme by t

red

ryp.

Redux: Overall Model for trp Operon